I can't comment on third party universal controllers, but the Fire Stick voice remote that came with my 2020 Firestick 4K controls Netflix (and BBC iPlayer and other services) through the Fire TV interface. It will also control TV on/off and volume. I don't know about source control because I don't CaTV, and content is only ever through Fire TV or terrestrial broadcast.
If you've got the older non-voice controller I can't comment, but if it is voice enabled, you've tried "asking Alexa", and adding Netflix and other apps to the Fire TV interface?
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